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Question: inverted pendulum question...

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Inverted pendulum question

Consider the electrical and mechanical dynamics of a DC motor (effect of L is neglected) that is loaded (without a gearbox reduction) by a rack&pinion gear set with pinion radius r and applies a rightwards pulling force F to an inverted pendulum system. We want to achieve a precise linear positioning of the cart with mass M (x) and angular positioning of the inverted pendulum with mass m (0). PLEASE show all mathematical steps to get partial credit. a) Derive the dynamic equations of the system including the inertia Jm) and In viscous friction (b) of the motor. Rotational inertia of the inverted pendulum is 1 =--. Motor voltage is V and motor shaft position is 0m b) Find the transfer function and c) Find the poles of both transfer functions, place the poles and comment on the stability of the plant.
J, 0.03995% 10-4 kgm b 0.9925 *10 Nms /rad K,-0.9925% 10-4 Vs / rad K,-3.2*10-3 Nm / A R 2.4Ω r-01m g-9.81m/s
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